in graphs from (a)-(d) we can find that the numbers of points with derivatives 0 are 2,0,1,3 respectively
and, in I-IV the numbers of points that interact with x-axis are 1,2,3,0, respectively
thus, from the above, we can get the answer:
(a) goes with ( II)
(b) goes with IV
(c) goes with I
(d) goes with III
extra explanation for (b)
(b) goes with IV, because the function in b is Increasing (f '>0), decreasing (f ' <0) and then increasing (f ' >0) .
The derivative along each segment is a constant (that's why the graph of its derivative has 3 horizontal line segments)
